A Liouville-Type Theorem for an Elliptic Equation with Superquadratic Growth in the Gradient
Keyword(s):
AbstractWe consider the elliptic equation {-\Delta u=u^{q}|\nabla u|^{p}} in {\mathbb{R}^{n}} for any {p>2} and {q>0}. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in [2], where the case {0<p<2} is considered. Some extensions to elliptic systems are also given.
2014 ◽
Vol 34
(11)
◽
pp. 4947-4966
◽
1994 ◽
pp. 405-415
◽
2017 ◽
Vol 63
(12)
◽
pp. 1704-1720
◽
2019 ◽
Vol 21
(01)
◽
pp. 1750069
◽
2000 ◽
Vol 244
(1)
◽
pp. 1-9
◽
Keyword(s):
2012 ◽
Vol 387
(1)
◽
pp. 153-165
◽
Keyword(s):
2007 ◽
Vol 326
(1)
◽
pp. 677-690
◽
2013 ◽
Vol 405
(2)
◽
pp. 608-617
2014 ◽
Vol 34
(9)
◽
pp. 3317-3339
◽
Keyword(s):